Talk by Francis Bach - Second Order Strikes Back - Globally convergent Newton methods for ill-conditioned generalized self-concordant Losses

We will study large-scale convex optimization algorithms based on the Newton method applied to regularized generalized self-concordant losses, which include logistic regression and softmax regression. We first prove that our new simple scheme based on a sequence of problems with decreasing regularization parameters is provably globally convergent, that this convergence is linear with a constant factor which scales only logarithmically with the condition number. In the parametric setting, we obtain an algorithm with the same scaling than regular first-order methods but with an improved behavior, in particular in ill-conditioned problems. Second, in the non parametric machine learning setting, we provide an explicit algorithm combining the previous scheme with Nyström projection techniques, and prove that it achieves optimal generalization bounds with a time complexity of order O(n\sqrt{n}), a memory complexity of order O(n) and no dependence on the condition number, generalizing the results known for least-squares regression (Joint work with Ulysse Marteau-Ferey and Alessandro Rudi, https://arxiv.org/abs/1907.01771).